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Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively. [9]: 160
A point in 4-dimensional space with Cartesian coordinates (u, x, y, z) may be represented by a quaternion P = u + xi + yj + zk. A left-isoclinic rotation is represented by left-multiplication by a unit quaternion Q L = a + bi + cj + dk .
A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system [8]) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning ...
A unit tesseract has side length 1, and is typically taken as the basic unit for hypervolume in 4-dimensional space. The unit tesseract in a Cartesian coordinate system for 4-dimensional space has two opposite vertices at coordinates [0, 0, 0, 0] and [1, 1, 1, 1], and other vertices with coordinates at all possible combinations of 0 s and 1 s.
In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of coordinate planes. Similarly, coordinate hypersurfaces are the (n − 1)-dimensional spaces resulting from fixing a single coordinate of an n-dimensional coordinate system. [14]
Where v is velocity, x, y, and z are Cartesian coordinates in 3-dimensional space, c is the constant representing the universal speed limit, and t is time, the four-dimensional vector v = (ct, x, y, z) = (ct, r) is classified according to the sign of c 2 t 2 − r 2. A vector is timelike if c 2 t 2 > r 2, spacelike if c 2 t 2 < r 2, and null or ...
Each event can be labeled by four numbers: a time coordinate and three space coordinates; thus spacetime is a four-dimensional space. The mathematical term for spacetime is a four-dimensional manifold (a topological space that locally resembles Euclidean space near each point). The concept may be applied as well to a higher-dimensional space.
The Rodrigues vector (sometimes called the Gibbs vector, with coordinates called Rodrigues parameters) [3] [4] can be expressed in terms of the axis and angle of the rotation as follows: = ^ This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from a 3-sphere onto the 3-dimensional pure ...